SPACES OF ABSOLUTELY SUMMING POLYNOMIALS

SPACES OF ABSOLUTELY SUMMING POLYNOMIALS

J. A. BARBOSA, G. BOTELHO, D. DINIZ and D. PELLEGRINO^∗

Abstract

This paper has a twofold purpose: to prove a much more general Dvoretzky-Rogers type theorem for absolutely summing polynomials and to introduce a more convenient norm on the space of everywhere summing polynomials.

1. Introduction

Since Pietsch [16], several nonlinear generalizations of absolutely summing operators have been investigated. Multilinear mappings/polynomials which are absolutely summing at a given point – and also everywhere – were introduced by M. Matos [9] and developed in [3], [7], [13], [14].

It is known that a Dvoretzky-Rogers-like theorem holds for everywhere summing polynomials (see [9]) but does not hold for summing polynomials (at the origin), so it is natural to ask whether or not such a theorem holds for polynomials which are absolutely summing at a pointa = 0. Proving in Section 3 a Dvoretzky-Rogers type theorem for absolutely summing polyno- mials at a given pointa =0, we provide a substantial improvement of Matos’

Dvoretzky-Rogers type theorem [9]. We also prove that summability at any point implies summability at the origin.

The norm that has been used in the space of everywhere summing polyno- mials (defined in [9]) has two inconvenients: (i) it is not a normalized ideal norm, in the sense that the everywhere summing norm of the polynomial x−→xⁿ, x∈K=scalar field, is not always equal to 1; (ii) it makes compu- tations quite difficult. In Section 4 we introduce another norm which happens to be equivalent to the original one and repairs the aforementioned inconvenients.

The multilinear case is also investigated.

2. Background and notation

Recall that, ifEandF are Banach spaces overK =RorCandp ≥q ≥1, a continuous linear operator u : E −→ F is absolutely (p;q)-summing

∗The second and fourth named authors are partially supported by IM-AGIMB. The fourth named author is also supported by CNPq/Fapesq.

Received March 14, 2006.

(or(p;q)-summing) if(u(xj))_j=^∞1is absolutelyp-summable inF whenever (xj)_j=^∞1 is weakly q-summable inE. For the theory of absolutely summing operators we refer to the book by Diestel-Jarchow-Tonge [4].

The multilinear theory of absolutely summing operators was introduced by Pietsch [16] and has been developed by several authors. There are various natural possible generalizations of the linear concept of absolute summability to polynomial/multilinear mappings (see [1], [5], [7], [10], [15]). If uis a linear operator, to estimate(u(a+xj)−u(a))_j=^∞1is the same as to estimate (u(xj))_j=^∞1. However, for polynomials, in general,P (a+x)=P (a)+P (x), as well as for multilinear mappings and hence, in the nonlinear case it makes sense to study absolute summabilitily with respect to a pointa=0. This idea is credited to Richard Aron, appeared for the first time in M. Matos [8] and was developed in [9] and in the doctoral thesis [12] of the fourth named author under supervision of Professor M. Matos.

As usual, the Banach space of all continuousn-homogeneous polynomials fromEintoF, with the sup norm, is represented byP(ⁿE;F ). The sequence spacesp(E)and^u_p(E)are defined by:

p(E)=

(xj)_j^∞₌1∈E^N; (xj)_j=^∞1p:= ∞

j=1

xj^p _p¹

<∞

,

^w_p(E)=

(xj)_j=^∞1∈E^N; (xj)_j=^∞1w,p := sup

ϕ∈BE

∞ j=1

|ϕ(xj)|^p _p¹

<∞ and lim

k→∞(xj)_j=k^∞ w,p =0

.

A polynomialP ∈P(ⁿE;F )is(p;q)-summing ata∈Eif

P (a+xj)− P (a)_∞

j=1 ∈p(F )for every(xj)_j=^∞1 ∈ ^u_q(E). It is not hard to prove that the class of all n-homogeneous polynomials from Einto F that are absolutely summing at a given point is a subspace ofP(ⁿE;F ). The space formed by the n-homogeneous polynomials that are(p;q)summing ata∈Ewill be denoted byP_as(p;q)^(a) (ⁿE;F ). Then-homogeneous polynomials that are(p;q)-summing ata = 0 will be simply called (p;q)-summing and the vector space of all (p;q)-summingn-homogeneous polynomials fromE intoF is represented byPas(p;q)(ⁿE;F ).

The space composed by then-homogeneous polynomials that are(p;q)- summing at every point is denoted byP_as(p;q)^ev (ⁿE;F ). Note that

P_as(p;q)^ev (ⁿE;F )=

a∈E

P_as(p;q)^(a) (ⁿE;F ).

IfP ∈ P_as(p;q)^ev (ⁿE;F ) we say that P is everywhere (p;q)-summing. The space of all continuousn-linear mappings from E1× · · · ×En intoF (with the sup norm) is denoted byL(E¹, . . . , En;F ) (L(ⁿE;F ) ifE¹ = · · · = En =E). We say thatT ∈L(E¹, . . . , En;F )is(p;q¹, . . . , qn)-summing at a=(a¹, . . . , an)∈E¹× · · · ×Enif

T (a1+x_j⁽¹⁾, . . . , an+x_j⁽ⁿ⁾)−T (a1, . . . , an)_∞

j=1∈p(F )

for every(x_j^(r))_j=^∞1 ∈ ^u_qr(Er), r = 1, . . . , n. As it happens for polynomi- als, it is easy to verify that the class of all n-linear mappings from E1×

· · · ×En into F which are (p;q¹, . . . , qn)-summing at a, represented by L_as(p;q^(a) _1,...,qn)(E¹, . . . , En;F ), is a subspace ofL(E¹, . . . , En;F ). The space formed by the n-linear mappings from E¹ × · · · × En into F which are (p;q1, . . . , qn)-summing at every point is denoted byL_as(p;q^ev _1,...,qn)(E1, . . . , En;F ). IfT ∈ L_as(p;q^ev _1,...,qn)(E1, . . . , En;F ) we say that T is everywhere (p;q¹, . . . , qn)-summing. Then-linear mappings that are(p;q¹, . . . ,qn)-sum- ming ata =0 will be simply called(p;q¹, . . . , qn)-summing and the vector space of all(p;q¹, . . . , qn)-summingn-linear mappings fromE¹× · · · ×En

intoF is represented byLas(p;q1,...,qn)(E1, . . . , En;F ).

Ifp = q = q1 = · · · = qn, instead of(p;p)or(p;p, . . . , p)-summing we say that the mapping isp-summing. In this case we write P_as,p^(a) (ⁿE;F ), Pas,p(ⁿE;F )andP_as,p^ev (ⁿE;F )for polynomials, and the adaptations for mul- tilinear mappings are obvious.

Nachbin’s concept of holomorphy type [11] was generalized in a natural way in [3] in the following fashion: aglobal holomorphy typePH is a subclass of the class of all continuous homogeneous polynomials between Banach spaces such that for every naturalnand every Banach spacesEandF, the component PH(ⁿE;F ) := P(ⁿE;F )∩PH is a linear subspace ofP(ⁿE;F )which is a Banach space when endowed with a norm denoted byP → PH, and

(i) PH(⁰E;F )=F, as a normed linear space for allEandF.

(ii) There isσ ≥ 1 such that for every Banach spaces E and F, n ∈ N, k ≤n,a∈EandP ∈PH(ⁿE;F ),dˆ^kP (a)∈PH(^kE;F )and

1

k!dˆ^kP (a) H

≤σⁿPHa^n−k,

wheredˆ^kP (a)is thek-th differential ofP ata(see [6], [11]).

3. Dvoretzky-Rogers type theorems

Two questions are treated in this section. The first question concerns a very useful result in the theory of summing linear operators, which happens to be a

weak version of the celebrated Dvoretzky-Rogers Theorem and asserts that if p≥1 andEis a Banach space, then

Eis finite dimensional⇐⇒Las,p(E;E)=L(E;E).

For polynomials and multilinear mappings, Matos [9] proved that ifn >1 and p≥1, then

Eis finite dimensional⇐⇒P_as,p^ev (ⁿE;E)=P(ⁿE;E)

⇐⇒L_as,p^ev (ⁿE;E)=L(ⁿE;E).

On the other hand, for polynomials/multilinear mappings summing at the origin this result is not valid in general: for example, from [2, Theorems 2.2 and 2.5]

we know thatPas,1(ⁿE;E) = P(ⁿE;E) andLas,1(ⁿE;E) = L(ⁿE;E)for everyn≥2 and every spaceEof cotype 2. The question is obvious: are there results of this type for polynomials and multilinear mappings summing at a pointa=0?

The second question arises from the well known fact that summability at the origin does not imply summability at a pointa = 0 in general (see [9, Example 3.2]). Again the question is obvious: is it true that summability at some pointa =0 implies summability at the origin?

We solve these two questions in the affirmative. The multilinear and poly- nomial cases demand different reasonings.

Multilinear case

We start by showing some connections betweenL_as(p;q)^(a) andL_as(p;q)^(b) fora= b. Some terminology is welcome. GivenT ∈ L(E¹, . . . , En;F ) and a = (a1, . . . , an)∈ E1× · · · ×En, we denote byTa1 the(n−1)-linear mapping fromE2× · · · ×EnintoF given by

Ta1(x², . . . , xn)=T (a¹, x², . . . , xn).

Analogously we define the(n−1)-linear mappingsTa2, . . . , Tan, the(n−2)- linear mappings Ta¹a² = T (a¹, a²,·, . . . ,·), . . . , Tan−1an = T (·, . . . ,·, an−1, an)and the linear mappings Ta¹,...,an−1 = T (a1, . . . , an−1,·), . . . , Ta²,...,an = T (·, a2, . . . , an).

Proposition 3.1. Let a = (a¹, . . . , an) ∈ E¹ × · · · × En and T ∈ L_as(p;q^(a) _1,...,qn)(E¹, . . . , En;F ). Then:

(a)Taj1,...,ajris(p;qk¹, . . . qks)-summing at the origin whenever{1, . . . , n} = {j1, . . . , jr} ∪ {k1, . . . , ks},k1≤. . .≤ksand{j1, . . . , jr} ∩ {k1, . . . , ks} = ∅. (b)T ∈ L_as(p;q^(b) _1,...,qn)(E1, . . . , En;F ) for everyb ∈ {(λ1a1, . . . , λnan);

λj ∈K, j =1, . . . , n}.

So, the set of all pointsbsuch thatT is(p;q1, . . . , pn)-summing atbcontains a linear subspace ofE1×· · ·×En. In particular,T is(p;q1, . . . , qn)-summing at the origin.

Proof. (a) For the linear operatorTa1...an−1 it is enough to observe that Ta1...an−1

x_j⁽ⁿ⁾

=T

a1+0, a2+0, . . . , an−1+0, an+x_j⁽ⁿ⁾

−T (a1, a2, . . . , an).

The cases ofTa1...an−2an, . . . , Ta2...an are analogous. For the bilinear mapping Ta¹...an−2, observe that

Ta¹...an−2

x_j⁽ⁿ⁻¹⁾, x_j⁽ⁿ⁾

= T

a¹+0, a²+0, . . . , an−2+0, an−1+x_j⁽ⁿ⁻¹⁾, an+x_j⁽ⁿ⁾

−T (a¹, . . . , an)

−T

a¹, a², . . . , an−1, x_j⁽ⁿ⁾

−T

a1, a2, . . . , an−2, x_j⁽ⁿ⁻¹⁾, an

= T

a1+0, a2+0, . . . , an−2+0, an−1+x_j⁽ⁿ⁻¹⁾, an+x_j⁽ⁿ⁾

−T (a¹, . . . , an)

−Ta1,...,an−1

x_j⁽ⁿ⁾

−Ta1,...,an−2an

x_j⁽ⁿ⁻¹⁾ . T is(p;q¹, . . . , qn)-summing ata by assumption and by the previous case we also know thatTa1,...,an−1 is(p;qn)-summing andTa1,...,an−2an is(p;qn−1)- summing, so it follows thatTa¹...an−2 is(p;qn−1, qn)-summing at the origin.

The other cases of bilinear mappings are analogous. Proceeding in this line, the proof can be completed.

(b) Letb= (λ1a1, . . . , λnan). Ifλj =0 for everyj, it suffices to observe that

∞ j=1

T

λ¹a¹+x_j⁽¹⁾, . . . , λnan+x_j⁽ⁿ⁾

−T (λ¹a¹, . . . , λnan)^p1_p

= ∞

j=1

T

λ1a1+ λ¹

λ1x_j⁽¹⁾, . . . , λnan+ λn

λnx_j⁽ⁿ⁾

−T (λ1a1, . . . , λnan) ^p1

p

=λ1. . . λn

∞ j=1

T

a1+ 1

λ¹x_j⁽¹⁾, . . . , an+ 1 λnx_j⁽ⁿ⁾

−T (a1, . . . , an) ^p_p¹

.

Now we use (a) to deal with the case in whichλj =0 for somej. The casen=3 illustrates the reasoning:T is(p;q1, q2, q3)-summing ata = (a1, a2, a3)by assumption, and from (a) we know that, at the origin, T is (p;q1, q2, q3)- summing,Ta1is(p;q², q³)-summing,Ta2is(p;q¹, q³)-summing,Ta3is(p;q¹,

q2)-summing,Ta1a2is(p;q3)-summing,Ta1a3is(p;q2)-summing andTa2a3is (p;q1)-summing.

•Caseλ¹=0,λ²=0 andλ³=0: follows from T (λ1a1+xj, λ2a2+yj, zj)−T (λ1a1, λ2a2,0)

=λ1λ2

T

a1+ xj

λ1, a2+ yj

λ2, zj

−T (a1, a2,0)

=λ¹λ²

T (a¹, a², zj)+T xj

λ¹, a², zj

+T

a1, yj

λ², zj

+T

xj

λ¹,yj

λ², zj

=λ¹λ²

Ta¹a²(zj)+Ta²

xj

λ1, zj

+Ta¹

yj

λ2, zj

+T

xj

λ1,yj

λ2, zj

.

•Casesλ¹=0,λ²=0,λ³=0 andλ¹=0,λ²=0,λ³=0 are analogous.

•Caseλ¹=0,λ²=λ³=0: follows from T (λ¹a¹+xj, yj, zj)−T (λ¹a¹,0,0)=λ¹

T

a¹+ xj

λ1, yj, zj

=λ1

T

a1, yj, zj

+T

xj

λ¹, yj, zj

.

•Casesλ²=0,λ¹=λ³=0 andλ³=0,λ²=λ¹=0 are analogous.

• Case λ1 = λ2 = λ3 = 0: we already know that T is(p;q1, q2, q3)- summing at the origin.

The following result is a significant improvement of Matos’ Dvoretzky- Rogers type theorem for multilinear mappings:

Theorem3.2. LetEbe a Banach space,n≥2andp≥1. The following assertions are equivalent:

(a) Eis infinite-dimensional.

(b) L_as,p^(a) (ⁿE;E)=L(ⁿE;E)for everya=(a1, . . . , an)∈Eⁿwith either ai =0for everyiorai =0for only onei.

(c) L_as,p^(a) (ⁿE;E)=L(ⁿE;E)for somea=(a¹, . . . , an)∈Eⁿwith either ai =0for everyiorai =0for only onei.

Proof. Since (b)⇒(c) is obvious and (c)⇒(a) is a direct consequence of [9, Theorem 6.3], we just have to prove (a)⇒(b): leta=(a1, . . . , an)∈Eⁿ with eitherai =0 for everyiorai =0 for only onei. We can fixk ∈ {1, . . . , n}

such thatai = 0 for every i = k. For each i = k choose ϕi ∈ E so that ϕi(ai)=1 and defineT ∈L(ⁿE;E)by

T (x¹, . . . , xn)=ϕ¹(x¹)· · ·ϕk−1(xk−1)ϕk+1(xk+1)· · ·ϕn(xn)xk. SinceTa¹...ak−1ak+1...an(x) = T (a1, . . . , ak−1, x, ak+1, . . . , an) = x for every x ∈ E, we have thatTa1...ak−1ak+1...an is notp-summing. From Proposition 3.1 it follows thatT is notp-summing ata.

From Proposition 3.1 we know that L_as,p^(a) (ⁿE;E) = L(ⁿE;E) ⇒ Las,p(ⁿE;E) = L(ⁿE;E). It is interesting to point out that Theorem 3.2 guarantees that much more holds in the bilinear case:

Corollary 3.3. Let E be an infinite-dimensional Banach space, a = (a1, . . . , an) ∈ Eⁿ, n ≥ 2 andp ≥ 1. IfL_as,p^(a)(ⁿE;E) = L(ⁿE;E), then card{i : ai = 0} ≥ 2. In particular, ifL_as,p^(a) (²E;E) = L(²E;E)thenais the origin.

Remark 3.4. The conditionai = 0 for everyi or ai = 0 for only one i is essential in Theorem 3.2: for example, it is not difficult to check that L_as,^(a)1(ⁿ¹;¹)= L(ⁿ¹;¹)for everya = (x,0,0, . . . ,0)with 0 = x ∈ ¹ and everyn≥3.

Polynomial case

The theory of summing polynomials at a given point has some specific tech- nical difficulties and deserves a precise examination. Despite the results we obtain for polynomials are analogous to the multilinear ones, the proofs of the multilinear results cannot be adapted to polynomials. For example, a polyno- mial version of Proposition 3.1 cannot be obtained following the lines of its proof. Such an adaptation would prove that ifP :E−→Fis(p;q)-summing ata ∈E,a =0, thenP is(p;q)-summing at everyλa,λ=0. Indeed, this implication follows from

P (λa+xj)−P (λa)=P

λa+ λ λxj

−P (λa)

=λⁿ

P

a+ 1 λxj

−P (a)

.

But we need more: we want to prove that ifP is(p;q)-summing at a = 0, thenP is(p;q)-summing at the origin. ByPˇ we mean the unique symmetric continuousn-linear mapping associated to then-homogeneous polynomialP.

Proposition3.5. LetP ∈P(ⁿE;F )anda ∈E.P is(p;q)-summing at aif and only ifPˇ is(p;q, . . . , q)-summing at(a, . . . , a)∈Eⁿ.

Proof. Using the polarization formula, the casea = 0 is immediate. We can supposea =0. Note that ifPˇ is(p;q, . . . , q)-summing at(a, . . . , a)it is plain thatP is(p;q)-summing ata. The proof of the other implication is divided in two cases:nodd andneven.

•First case:nis odd. In this case the polarization formula is decisive:

(3.1) n!2ⁿPˇ

a+x_j⁽¹⁾, . . . , a+x_j⁽ⁿ⁾

− ˇP (a, . . . , a)

=

εi=±1

ε¹· · ·εnP

ε¹(a+x_j⁽¹⁾)+ · · · +εn(a+x_j⁽ⁿ⁾)

−

εi=±1

ε1· · ·εnP (ε1a+ · · · +εna)

=

εi=±1

ε1· · ·εn

P

(ε1a+ · · · +εna)+(ε1x_j⁽¹⁾+ · · · +εnx_j⁽ⁿ⁾)

−P (ε¹a+ · · · +εna) .

Sincenis odd,(ε¹+ · · · +εn)=0.P is(p;q)-summing ataby assumption, so according to what we did above it follows thatP is(p;q)-summing at each (ε1a+ · · · +εna). Thus (3.1) yields thatPˇ is(p;q)-summing at(a, . . . , a).

• Second case:n is even. Chooseϕ ∈ E so that ϕ(a) = 1 and define Q∈ P(ⁿ⁺¹E;F )byQ(x) = ϕ(x)P (x). Using thatP ∈ P_as(p;q)^(a) (ⁿE;F ), it is easy to check that Qis(p;q)-summing ata. But(n+1)is odd, so the previous case can be invoked in order to conclude thatQˇ is(p;q)summing at (a, . . . , a). SinceQˇaandϕare(p;q)-summing at the origin (the case ofQˇa

follows from Proposition 3.1), from

Qˇa(x, . . . , x)= ˇQ(a, x, . . . , x)= (n−1)

n P (a, x, . . . , x)ϕ(x)ˇ + 1 nP (x) we conclude that P is (p;q)-summing at the origin as well. Now, the po- larization formula can be invoked as in (3.1) in order to conclude that Pˇ is (p;q)-summing at(a, . . . , a)and the proof is done.

Applying Proposition 3.1 once and Proposition 3.5 twice we have:

Corollary3.6.LetP ∈P(ⁿE;F )be(p;q)-summing ata∈E. ThenP is(p;q)-summing atλafor everyλ∈K. In particular,P is(p;q)-summing at the origin.

Now we obtain the Dvoretzky-Rogers type theorem for polynomials sum- ming at a pointa=0.

Theorem3.7.LetEbe a Banach space,n≥2andp≥1. The following assertions are equivalent:

(a) Eis infinite-dimensional.

(b) P_as,p^(a) (ⁿE;E)=P(ⁿE;E)for everya∈E,a=0.

(c) P_as,p^(a) (ⁿE;E)=P(ⁿE;E)for somea∈E,a=0.

Proof. As in the proof of Theorem 3.2, we just have to prove (a)⇒(b):

leta∈E,a =0. Chooseϕ∈E so thatϕ(a) =1 and defineP ∈P(ⁿE;E) byP (x)= ϕ(x)ⁿ⁻¹x. Assume thatP isp-summing ata. By Proposition 3.5 we have that Pˇ is p-summing at (a, . . . , a). Defining Pa ∈ L(E;E) by Pa(x)= ˇP (a, . . . , a, x), from

Pa(x)= ˇP (a+0, . . . , a+0, a+x)− ˇP (a, . . . , a) for every x∈E, we conclude thatPaisp-summing. From

Pa(x)= (n−1)

n ϕ(x)a+ 1

nx for every x ∈E,

it follows that the identity operator onE isp-summing. This contradiction completes the proof.

4. Norms on spaces of everywhere summing polynomials

In order to define a norm on the space P_as(p;q)^ev (ⁿE;F ) of everywhere (p;q)-summing polynomials, Matos [9], in a clever argument, for eachP ∈ P_as(p;q)^ev (ⁿE;F )considered the polynomial

p;q(P ):^u_q(E)−→p(F );(xj)_j=^∞1−→(P (x1), (P (x1+xj)−P (x1))_j=^∞2) and showed that the the correspondenceP −→ p;q(P )defines a norm onP_as(p;q)^ev (ⁿE;F ). We shall denote this norm byP_ev⁽¹⁾_(p;q). Matos proved that this norm is complete and that (P_as(p;q)^ev ,·_ev⁽¹⁾_(p;q)) is a global holo- morphy type. Matos’ argument was recently adapted to multilinear mappings in [3] (henceforth we whall writeL_as(p;q)^ev instead ofLas(p;q,...,q)^ev ): givenT ∈ L_as(p,q)^ev (E¹, . . . , En;F ), consider the multilinear mappingξp;q(T ):^u_q(E¹)×

· · · ×^u_q(En)−→p(F )given by x_j⁽¹⁾_∞

j=1, . . . , x_j⁽ⁿ⁾_∞

j=1

−→

T (x1⁽¹⁾, . . . , x1⁽ⁿ⁾),

T (x1⁽¹⁾+x_j⁽¹⁾, . . . , x1⁽ⁿ⁾+x_j⁽ⁿ⁾)−T (x⁽1¹⁾, . . . , x1⁽ⁿ⁾)_∞

j=2

. In [3] it is proved that the correspondenceT −→ ξp,q(T )defines a complete norm onL_as(p;q)^ev (ⁿE;F ), which we shall denote byT_ev⁽¹⁾_(p;q). So, inP_as(p;q)^ev

another natural norm is defined by P_ev^{(I )}_(p;q) := ˇPev⁽¹⁾(p;q). In [3] it is shown that with this normP_as(p;q)^ev is also a global holomorphy type.

We will see that these ideal norms onP_as(p;q)^ev andL_as(p;q)^ev are non-normal- ized in general and present quite serious difficulties concerning computations, even for very simple mappings. Our aim in this section is to introduce normal- ized ideal norms onP_as(p;q)^ev andL_as(p;q)^ev which happen to be equivalent to the original norms and make computations quite easier.

Next two theorems are adaptations of Matos’ argument.

Theorem4.1.The following assertions are equivalent forT ∈L(E1, . . . , En;F ):

(a) T ∈L_as(p;q)^ev (E1, . . . , En;F ). (b) There existsCsuch that

∞ j=1

T (b¹+x_j⁽¹⁾, . . . , bn+x_j⁽ⁿ⁾)−T (b¹, . . . , bn)^p_p¹

≤C

b¹ +x_j⁽¹⁾_∞

j=1

w,q

. . .

bn +x_j⁽ⁿ⁾_∞

j=1

w,q

for every(b¹, . . . , bn)∈E¹× · · · ×Enand(x_j^(k))_j=^∞1∈^u_q(Ek), k=1, . . . , n. Moreover, the infimum of allC for which(b)holds defines a complete norm onL_as(p;q)^ev denoted by·_ev⁽²⁾_(p;q).

Proof. Since (b)⇒(a) is obvious we just have to prove (a)⇒(b): define Gk =Ek×^u_q(Ek),k=1, . . . , n, and consider then-linear mapping _p;q(T ): G1× · · · ×Gn−→p(F )given by

a1, x_j⁽¹⁾_∞

j=1

, . . . , an,

x_j⁽ⁿ⁾_∞

j=1

−→

T (a1+x_j⁽¹⁾, . . . , an+x_j⁽ⁿ⁾)−T (a1, . . . , an)_∞

j=1. Following the lines of the proofs of [3, Propositions 9.3 and 9.4] it can be proved that _p;q(T )is continuous and that the correspondenceT −→ p;q(T ):= T_ev⁽²⁾_(p;q)defines a complete norm onL_as(p;q)^ev (E¹, . . . , En;F ).

Theorem4.2.The following assertions are equivalent forP ∈P(ⁿE;F ): (a) P ∈P_as(p;q)^ev (ⁿE;F ).

(b) There existsCsuch that (4.1)

∞ j=1

P (a+xj)−P (a)^p_p¹

≤C

a +(xj)_j=^∞1

w,q

_n

for everya ∈ Eand(xj)_j=^∞1 ∈ ^u_q(E). Moreover, the infimum of allC for which(b)holds defines a complete norm onP_as(p;q)^ev (ⁿE;F )denoted by._ev⁽²⁾_(p;q).

Proof. Again we just have to prove (a)⇒(b): defineG=E×^u_q(E)and consider the polynomial

ηp;q(P ):G−→p(F );

a, (xj)_j=^∞1

−→

P (a+xj)−P (a)_∞

j=1. Following the lines of the proofs of [9, Theorem 7.2 and Proposition 7.4] it can be proved thatηp;q(P )is continuous and that the correspondenceP −→

ηp;q(P ):= P_ev⁽²⁾_(p;q)defines a complete norm onP_as(p;q)^ev (ⁿE;F ). We can also consider the norm on P_as(p,q)^ev defined by P_ev^(II)_(p;q) := ˇP_ev⁽²⁾_(p;q). So we have four norms on P_as(p,q)^ev , namely ·_ev⁽¹⁾_(p;q),

·_ev⁽²⁾_(p;q),·_ev^{(I )}_(p;q)and·_ev^(II)_(p;q). We will show that: (i) these four norms are distinct in general but equivalent; (ii) the ideal (P_as(p,q)^ev ,·_ev⁽²⁾_(p;q)) is normalized; (iii) the ideal (P_as(p,q)^ev ,·_ev⁽¹⁾_(p;q)) is non-normalized in gen- eral; (iv) the norm·_ev⁽²⁾_(p;q)is easier for computations; (v) these four norms makeP_as(p,q)^ev a global holomorphy type. In our opinion these facts show that

·_ev⁽²⁾_(p;q)is the most convenient norm onP_as(p,q)^ev and justify its introduction.

Multilinear case

Given n ∈ N, by An:Kⁿ −→ K we mean the canonical n-linear mapping given byAn(x1, . . . , xn)=x1· · ·xn. According to the usual axiomatization, a Banach ideal of multilinear mappings(M, · M)must satisfy the condition AnM =1 for everyn.

Proposition4.3. Letn∈N.

(a) An_ev⁽²⁾_(p;q)=1for everyp≥q ≥1.

(b) Anev⁽¹⁾(p;1)=1for everyp≥1.

(c) An_ev⁽¹⁾_(p;q) ≥ 2^q1∗, where ¹

q + _q¹∗ = 1, for every p ≥ q > 1. In particular,Anev⁽¹⁾(p;q)>1wheneverq >1.

(d) lim_n→∞An_ev⁽¹⁾_(p;q)= ∞for everyp≥q >1.

Proof. By definition it is obvious thatAn_ev⁽¹⁾_(p;q) ≥ Anas(p;q) = 1 andAnev⁽²⁾(p;q)≥ Anas(p;q)=1.

(a) We just have to prove thatAnev⁽²⁾(p;q) ≤ 1. The casen= 3 is illus- trative: givena1, a2, a3 ∈Kand(x_j¹), (x_j²), (x_j³)∈ q =^u_q(K), sincep ≥q

we have

∞

j=1

|A3(a1+x_j¹, a2+x_j², a3+x_j³)−A3(a1, a2, a3)|^p _p¹

=

∞

j=1

|a1a2x_j³+a1a3x_j²+a1x_j²x_j³+a2a3x_j¹+a2x_j¹x_j³+a3x_j¹x_j²+x_j¹x_j²x_j³|^p

1 p

≤ |a1a2|

∞

j=1

|x_j³|^{q 1}_q

+ |a1a3|

∞

j=1

|x_j²|^{q 1}_q

+ |a1|

∞

j=1

|x_j²x_j³|^{q 1}_q

+ |a2a3|

∞

j=1

|x_j¹|^{q 1}_q

+ |a2|

∞

j=1

|x_j¹x_j³|^{q 1}_q

+ |a3|

∞

j=1

|x_j¹x_j²|^{q 1}_q

+

∞

j=1

|x_j¹x_j²x_j³|^q _q¹

≤ |a1a2|

∞

j=1

|x_j³|^{q 1}_q

+ |a1a3|

∞

j=1

|x_j²|^{q 1}_q

+ |a1|

∞

j=1

|x_j²|^q

∞

j=1

|x_j³|^{q 1}_q

+ |a2a3|

∞

j=1

|x_j¹|^{q 1}_q

+ |a2|

∞

j=1

|x_j¹|^q

∞

j=1

|x_j³|^q _q¹

+ |a3|

∞

j=1

|x_j¹|^q

∞

j=1

|x_j²|^q _q¹

+

∞

j=1

|x_j¹|^q

∞

j=1

|x_j²|^q

∞

j=1

|x_j³|^{q 1}_q

=

|a1| +

∞

j=1

|x_j¹|^{q 1}_q

|a2| +

∞

j=1

|x_j²|^q _q¹

|a3| +

∞

j=1

|x_j³|^{q 1}_q

− |a1a2a3|

≤

|a1| + (x_j¹)q

|a2| + (x_j²)q

|a3| + (x_j³)q

=

|a1| + (x_j¹)w,q

|a2| + (x_j²)w,q

|a3| + (x_j³)w,q

proving thatA3_ev⁽²⁾_(p;q)≤1.

(b) In essence, the same argument of (a). Use thatp≥1 implies·_p ≤ ·1

and in the caseq =1, the last line of the above computation coincides with (a1, (x_j¹))

w,1·(a2, (x_j²))

w,1·(a3, (x_j³))

w,1. (c) We know that

(4.2)

|a1· · ·an|^p+ ∞ j=1

|(a1+x_j¹)· · ·(an+x_jⁿ)−a1· · ·an|^{p 1}

p

≤ An_ev⁽¹⁾_(p;q)

|a1|^q+^∞

j=1

|x_j¹|^q _q¹

· · ·

|an|^q+^∞

j=1

|x_jⁿ|^q _q¹

,

for every ak ∈ Kand (x_j^k)_j^∞₌1 ∈ q, k = 1, . . . , n. Choosing a1 = · · · =